Vector Algebra Video Lecture - Civil Engineering (CE)

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we are going to look at water vectors
a vector okay is this a vector R is this
a vector or some of you have been using
a notation with the arrow on top is this
a vector or is this a vector is this a
vector or is this a vector which of
these is a vector I have given five
choices which of these five choices is a
vector for mathematicians any of these
five choices represents a vector but for
mechanics purposes when you say a vector
we mean a directed line segment this is
what we mean by a vector okay this what
we mean by a vector this is for us a
vector that is directed line segment
okay it is a geometrical object
it is a geometrical object and in that
is a vector okay that in mechanic's we
can say this has the vector now let us
understand what do you mean by addition
of two vectors
say I add data line segments like this
another data in segments like that
and I say I'm going to add these two I
bring this data line segment such that I
move this parallely on to this okay
assume these two are parallel then this
is the resultant vector this is the
resultant vector see if I denote this by
a B I write it as a plus B to B equal to
C okay for us if you were to be if you
were to be written as a 1 a 2 a 3 and B
were to be written as B 1 B 2 B 3 then C
is a1 plus b1 a2 plus b2 a3 plus b3 is
this is a special case
we will see when we can write EAS a 1 a
2 a 3 B s B 1 B 2 B 3 but what we mean
by addition is the diagonal formed by a
parallelogram with a and B as the sides
of the parallelogram is what we mean by
addition of two vectors there are cases
where you can write D components of
vector D a and B as this and the
components of a JCL become this under
some special circumstances what they are
and how to when it becomes like this we
will see subsequently ok next we are
interested in seeing a scalar
multiplication plication of a vector
that is why our vector a which is that
line segment we want alpha times a what
it means is and if alpha is if alpha is
2 then this vector these are parallel
lines where the length has become - okay
this length was L this length has now
become 2l and the action remains the
same on the other hand if alpha were to
be minus 1 the direction reverses and
the length remains the same this is
remains CL by dr. nerse reversed and
this would be if this B vector this is
the B vector yeah okay that's a B vector
then that's what scalar multiplication
of a vector mean on the other hand a dot
product means of two vectors is denoted
by a dotted with B is equal to some
scalar value C this is not a vector
scalar value okay what it means is if I
have a vector a I have actor B that line
segment I project this vector onto this
and this length is denoted by a dot R
would be okay that length is noted by a
dotted by with B in other words in terms
of length of a and B this would be
mounted of a magnitude of B times cosine
of theta I theta is the angle between a
and B okay that's what a dot product
means here I am assuming that magnitude
of a here I assume and
to be equal to one so thats why its
production of beyond to a the length is
given by magnitude of a which is a start
with a okay I am sure you know all these
things already is just to recap on
certain basic operations of vector
algebra and how do we represent those
operations in this course okay next is
cross product will do not this by this
symbol in this course if I have two
vectors a B then C is defined as a cross
B okay C is defined as a cross B then C
would be perpendicular to this plane
containing a and B we say B sees a
vector which is perpendicular to this
plane hundred of C would be nothing but
area of the parallelogram
with vectors a and B as its sides okay
that is a complete this parallelogram
this area denotes the magnitude of C
okay so that's what across for our des
okay now we have to introduce two
symbols to be able to mathematically
represent the cross product and the dot
product okay
Delta IJ is equal to 1 Phi equal to J
and 0 Phi is not equal to J this is
called as Kronecker Delta okay
that is Delta 1 1 would be equal to
Delta 2 to be equal to Delta 3 3 would
be equal to 1 but Delta 1 2 will be
equal to Delta 2 1 equal to Delta 1 3
would be equal to Delta 3 1 be equal to
Delta 2 3 equal Delta 3 2 all this would
be 0 ok so this is called as a Kronecker
Delta this is used to mathematically use
for dot product
okay for a cross-product what we do is
we use the symbol epsilon ijk which is
called as the permutation symbol
which takes values zero if i equal to j
r j equal to k r k equal to i and one if
i j k go in a clockwise direction that
is one two and three go in a clockwise
direction that is epsilon 1 2 is equal
to epsilon 2 3 is equal to epsilon 3 1
all this would be equal to 1 okay and it
will be minus 1 if 1 2 & 3 are going in
a and clockwise direction that is
epsilon 1 3 is equal to epsilon 3 2 is
equal to epsilon 2 1 would be minus 1
note the order of index 1 2 and 2 1 our
different signs here okay this is called
as a permutation symbol this used for
cross product
okay we're disuse our disuse will
understand shortly okay now there is one
more project we have to see that is
called as the box product our scalar
triple product
okay what this tells us is use this is
if I add three vectors a dot B cross C
will be denoted by a comma B comma C
there's a combination of a cross-border
and a dot product and what this
represents is volume of a parallelepiped
with vectors ABC as its site so
basically if I have a vector B vector
and a C vector I form a parallelepiped
which is
this and the volume contained in this
parallelepiped is what this scalar
triple product gives us okay so we have
seen four products now a dot product a
cross product a scalar triple product
scalar multiplication and vector
addition okay so basically these are
some vector operations and we have seen
you will see that we will use this to a
skeletal product to find the volume
changes when the body undergoes
deformation will use the cross product
to determine the AR changes when body
undergoes a deformation and so on when
you look at the corresponding
deformation patterns of the body okay so
we will stop here for today's lecture
basically in today's class you have got
an introduction to linear algebra how do
you multiply two matrices and then we
got to know about indicial notation what
is called as a dummy index and a free
index and in the dummy index will repeat
only twice on one side or equal to sign
and a free index will appear on either
side of the equal to sign a dummy index
has to be sounds from one two three
whereas a free index need not be sung
from one two three okay and then we got
an introduction to vector algebra we saw
what a vector is for us a vector is a
data line segment okay this is a
geometrical object okay and then we saw
a sha what addition of two vectors mean
what scalar multiplication of two
vectors a vector means and then we saw
what a scalar multiplication are dot
product of a vector means two vectors
means and then we saw what the scalar
triple product means and then what a
vector product of two vectors means okay
so with this we'll conclude today's
lecture thank you
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